You need to finish your story missions to around level 20 and join a grand company. After that you get the quest to raise your own chocobo.
Well, you don't really raise one. You just earn enough seals to exchange them for a license, then you show the license to this guy and he gives you a chocobo. --- Currently playing: Final Fantasy XIV: ARR (Beta) (PS3) PSN: TetsuyaHikari / Youtube: GamingHeroTetsuya